Finite Group Databasebeta

← #78865

Subgroups — #78865

Order 384.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5825C211yesmaximal, characteristic
2C2×#951C211yesmaximal, characteristic
3C22×Dic12C211yesmaximal, characteristic, complemented
4#5516C211yesmaximal, characteristic, complemented
5C3×#416C211yesmaximal, characteristic
6#5102C211yesmaximal, characteristic, complemented
7#5004C211yesmaximal, characteristic, complemented
8#4795C214yesmaximal, complemented
9#4767C214yesmaximal, complemented
10#3044—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×Dic6C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C22×C3×C8C2211yescharacteristic
15C3×#117C2211yescharacteristic
16#951—24no—
17#944C2212yes—
18C2×#216C2212yes—
19#936C2212yes—
20#935—22no—
21C2×#214C2212yes—
22#926C2211yescharacteristic
23#924—22no—
24C2×#208C2211yescharacteristic
25C2×Dic12—24no—
26C2×Dic12C2214yes—
27#887C2214yes—
28#877—22no—
29C4×Dic6—22no—
30C3×#103C2212yes—
31C3×#102C2212yes—
32Dic3⋊C8C2214yes—
33#810C2214yes—
34C2×#137—31no—
35C22×Q16—31no—
36#536—31no—
37#519—31no—
38#464—34no—
39#462—34no—
40#424—31no—
41#419—31no—
42#416D311yescharacteristic
43C3×#41—22no—
44C3×#41—22no—
45C2×C3×Q8C2311yescharacteristic
46C2×C3×D4—21no—
47C2×C3×D4C2311yescharacteristic
48C22×C3×C4—21no—
49C22×C3×C4C2311yescharacteristic
50C22×Dic3—21no—
51C22×Dic3—21no—
52C2×Dic6—21no—
53C2×Dic6—22no—
54C2×Dic6—22no—
55C2×Dic6—22no—
56C2×Dic6—21no—
57C2×Dic6D414yes—
58C2×Dic6C2312yes—
59C2×Dic6C2312yes—
60C2×C3×C8—22no—
61C2×C3×C8C2312yes—
62C3×C4⋊C4C2312yes—
63C3×C22⋊C4—21no—
64C3×C42—21no—
65#217—24no—
66#217D412yes—
67#216—24no—
68#214—24no—
69C4⋊Dic3D412yes—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72C4×Dic3—24no—
73C4×Dic3C2312yes—
74#208—22no—
75C2×C3⋊C8C2312yes—
76Dic12—28no—
77C2×#41C2×D311yescharacteristic
78C22×Q8—31no—
79C22×Q8—31no—
80#137—64no—
81C2×Q16—32no—
82C2×Q16—34no—
83C2×Q16—64no—
84C2×SD16—32no—
85C2×M16—31no—
86C22×C8C2×D311yescharacteristic
87C4⋊Q8—32no—
88#124—32no—
89#123—62no—
90C22⋊Q8—62no—
91C4×Q8—34no—
92C4×Q8—62no—
93#117—31no—
94#117—62no—
95#117C2×D311yescharacteristic
96C4⋊C8—34no—
97#103—34no—
98#103C2×D312yes—
99#102C2×D312yes—
100C23×C3—21no—
101C23×C3C2×D411yescharacteristic
102C3×Q8—21no—
103C3×Q8—21no—
104C3×D4—21no—
105C3×D4—21no—
106C3×D4—42no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—42no—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D412yes—
112C2×C3×C4#4112yes—
113C2×C3×C4C2411yescharacteristic
114C2×Dic3—24no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—42no—
118C2×Dic3C2×D414yes—
119Dic6—24no—
120Dic6—22no—
121Dic6—22no—
122Dic6—22no—
123Dic6—42no—
124Dic6C2×D414yes—
125C3×C8—22no—
126C3⋊C8—22no—
127#41—22no—
128#41—22no—
129C2×Q8—34no—
130C2×Q8—32no—
131C2×Q8—32no—
132C2×Q8—62no—
133C2×Q8—61no—
134C2×Q8—61no—
135C2×Q8—62no—
136C2×Q8—62no—
137C2×Q8C22×D311yescharacteristic
138C2×D4—21no—
139C2×D4C22×D311yescharacteristic
140C22×C4—21no—
141C22×C4—61no—
142C22×C4—61no—
143C22×C4C22×D311yescharacteristic
144Q16—68no—
145Q16—64no—
146SD16—64no—
147M16—62no—
148C2×C8—22no—
149C2×C8—32no—
150C2×C8C22×D312yes—
151C4⋊C4—32no—
152C4⋊C4—64no—
153C4⋊C4—64no—
154C4⋊C4C22×D312yes—
155C22⋊C4—21no—
156C22⋊C4—32no—
157C22⋊C4—64no—
158C42—21no—
159C42—32no—
160C42—64no—
161C22×C3—22no—
162C22×C3—41no—
163C22×C3—42no—
164C22×C3#12112yes—
165C22×C3C22×D411yescharacteristic
166C3×C4—41no—
167C3×C4—42no—
168C3×C4#12112yes—
169C3×C4C22×D411yescharacteristic
170C3×C4C2×#4111yescharacteristic
171Dic3—24no—
172Dic3—24no—
173Dic3—42no—
174C23—21no—
175C23D3×D411yescharacteristic
176Q8—21no—
177Q8—21no—
178Q8—34no—
179Q8—64no—
180Q8—62no—
181Q8—62no—
182Q8—62no—
183Q8—122no—
184D4—21no—
185D4—21no—
186D4—42no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—34no—
190C2×C4—42no—
191C2×C4—64no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—122no—
195C2×C4#23512yes—
196C2×C4D3×D411yescharacteristic
197C2×C4D3×D412yes—
198C2×C4C23×D311yescharacteristic
199C8—22no—
200C8—62no—
201C2×C3—22no—
202C2×C3—81no—
203C2×C3C2×#12111yescharacteristic
204C2×C3#57812yes—
205C22—22no—
206C22—41no—
207C22—42no—
208C22#88412yes—
209C22C2×D3×D411yescharacteristic
210C4—41no—
211C4—42no—
212C4—64no—
213C4—64no—
214C4—122no—
215C4#88412yes—
216C4C2×#23511yescharacteristic
217C4C2×D3×D411yescharacteristic
218C3#304411yescharacteristic, complemented, Sylow
219C2—22no—
220C2—81no—
221C2C2×#88411yescharacteristic
222C2#577112yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#416C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#416D3
hypocenterC3#3044
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C22×Dic12C2semidirect
4. #5516C2semidirect
6. #5102C2semidirect
7. #5004C2semidirect
8. #4795C2semidirect
9. #4767C2semidirect
218. C3#3044semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#58252
C2×#9512
C22×Dic122
#55162
C3×#4162
#51022
#50042
#47952
#47672
#30443